Recap Datalog Datalog Syntax Datalog Semantics. Logic: Datalog. CPSC 322 Logic 6. Textbook Logic: Datalog CPSC 322 Logic 6, Slide 1

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1 Logic: Datalog CPSC 322 Logic 6 Textbook 12.2 Logic: Datalog CPSC 322 Logic 6, Slide 1

2 Lecture Overview 1 Recap 2 Datalog 3 Datalog Syntax 4 Datalog Semantics Logic: Datalog CPSC 322 Logic 6, Slide 2

3 Top-down Ground Proof Procedure Idea: search backward from a query to determine if it is a logical consequence of KB. An answer clause is of the form: yes a 1 a 2... a m The SLD Resolution of this answer clause on atom a i with the clause: is the answer clause a i b 1... b p yes a 1 a i 1 b 1 b p a i+1 a m. Logic: Datalog CPSC 322 Logic 6, Slide 3

4 Derivations An answer is an answer clause with m = 0. That is, it is the answer clause yes. A derivation of query?q 1... q k from KB is a sequence of answer clauses γ 0, γ 1,..., γ n such that γ 0 is the answer clause yes q 1... q k, γ i is obtained by resolving γ i 1 with a clause in KB, and γ n is an answer. Logic: Datalog CPSC 322 Logic 6, Slide 4

5 Top-down definite clause interpreter To solve the query?q 1... q k : ac := yes q 1... q k repeat select atom a i from the body of ac; choose clause C from KB with a i as head; replace a i in the body of ac by the body of C until ac is an answer. Recall: Don t-care nondeterminism If one selection doesn t lead to a solution, there is no point trying other alternatives. select Don t-know nondeterminism If one choice doesn t lead to a solution, other choices may. choose Logic: Datalog CPSC 322 Logic 6, Slide 5

6 Lecture Overview 1 Recap 2 Datalog 3 Datalog Syntax 4 Datalog Semantics Logic: Datalog CPSC 322 Logic 6, Slide 6

7 Objects and Relations It is useful to view the world as consisting of objects and relationships between these objects. Often the propositions we spoke about before can be condensed into a much smaller number of propositions if they are allowed to express relationships between objects and/or functions of objects. Thus, reasoning in terms of objects and relationships can be simpler than reasoning in terms of features, as you can express more general knowledge using logical variables. Logic: Datalog CPSC 322 Logic 6, Slide 7

8 Using an RRS 1 Begin with a task domain. 2 Distinguish those objects you want to talk about. 3 Determine what relationships you want to represent. 4 Choose symbols in the computer to denote objects and relations. 5 Tell the system knowledge about the domain. 6 Ask the system questions. Logic: Datalog CPSC 322 Logic 6, Slide 8

9 Example Domain for an RRS in(alan,r123). part_of(r123,cs_building). in(x,y) part_of(z,y) in(x,z). alan r123 r023 cs_building in(, ) part_of(, ) person( ) in(alan,cs_building) Logic: Datalog CPSC 322 Logic 6, Slide 9

10 Representational Assumptions of Datalog An agent s knowledge can be usefully described in terms of individuals and relations among individuals. An agent s knowledge base consists of definite and positive statements. The environment is static. There are only a finite number of individuals of interest in the domain. Each individual can be given a unique name. = Datalog Logic: Datalog CPSC 322 Logic 6, Slide 10

11 Lecture Overview 1 Recap 2 Datalog 3 Datalog Syntax 4 Datalog Semantics Logic: Datalog CPSC 322 Logic 6, Slide 11

12 Syntax of Datalog Definition (variable) A variable starts with upper-case letter. Definition (constant) A constant starts with lower-case letter or is a sequence of digits. Definition (term) A term is either a variable or a constant. Definition (predicate symbol) A predicate symbol starts with lower-case letter. Logic: Datalog CPSC 322 Logic 6, Slide 12

13 Syntax of Datalog (cont) Definition (atom) An atomic symbol (atom) is of the form p or p(t 1,..., t n ) where p is a predicate symbol and t i are terms. Definition (definite clause) A definite clause is either an atomic symbol (a fact) or of the form: }{{} a b 1 b }{{ m } head body where a and b i are atomic symbols. Definition (knowledge base) A knowledge base is a set of definite clauses. Logic: Datalog CPSC 322 Logic 6, Slide 13

14 Example Knowledge Base in(alan, R) teaches(alan, cs322) in(cs322, R). grandf ather(william, X) father(william, Y ) parent(y, X). slithy(toves) mimsy borogroves outgrabe(mome, Raths). Logic: Datalog CPSC 322 Logic 6, Slide 14

15 Lecture Overview 1 Recap 2 Datalog 3 Datalog Syntax 4 Datalog Semantics Logic: Datalog CPSC 322 Logic 6, Slide 15

16 Semantics: General Idea Recall: a semantics specifies the meaning of sentences in the language. ultimately, we want to be able to talk about which sentences are and which are false In propositional logic, all we needed to do in order to come up with an interpretation was to assign truth values to atoms For Datalog, an interpretation specifies: what objects (individuals) are in the world the correspondence between symbols in the computer and objects & relations in world which constants denote which individuals which predicate symbols denote which relations (and thus, along with the above, which sentences will be and which will be false) Logic: Datalog CPSC 322 Logic 6, Slide 16

17 Formal Semantics Definition (interpretation) An interpretation is a triple I = D, φ, π, where D, the domain, is a nonempty set. Elements of D are individuals. φ is a mapping that assigns to each constant an element of D. Constant c denotes individual φ(c). π is a mapping that assigns to each n-ary predicate symbol a relation: a function from D n into {TRUE, FALSE}. Logic: Datalog CPSC 322 Logic 6, Slide 17

18 Example Interpretation Constants: phone, pencil, telephone. Predicate Symbol: noisy (unary), lef t of (binary). D = {,, }. These are actually objects in the world, not symbols φ(phone) =, φ(pencil) =, φ(telephone) =. π(noisy): FALSE TRUE FALSE π(left of):, FALSE, TRUE, TRUE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE Logic: Datalog CPSC 322 Logic 6, Slide 18

19 Important points to note The domain D can contain real objects. (e.g., a person, a room, a course). D can t necessarily be stored in a computer. The constants do not have to match up one-to-one with members of the domain. Multiple constants can refer to the same object, and some objects can have no constants that refer to them. π(p) specifies whether the relation denoted by the n-ary predicate symbol p is or false for each n-tuple of individuals. If predicate symbol p has no arguments, then π(p) is either TRUE or FALSE. this was the situation in propositional logic Logic: Datalog CPSC 322 Logic 6, Slide 19

20 Truth in an interpretation Definition (truth in an interpretation) A constant c denotes in I the individual φ(c). Ground (variable-free) atom p(t 1,..., t n ) is in interpretation I if π(p)(t 1,..., t n) = TRUE, where t i denotes t i in interpretation I and false in interpretation I if π(p)(t 1,..., t n) = FALSE. Ground clause h b 1... b m is false in interpretation I if h is false in I and each b i is in I, and item in interpretation I otherwise. A knowledge base, KB, is in interpretation I if and only if every clause in KB is in I. Notice that truth values are only associated with predicates (atomic symbols; clauses), not variables and constants! Logic: Datalog CPSC 322 Logic 6, Slide 20

21 Example Truths In the interpretation given before: noisy(phone) Logic: Datalog CPSC 322 Logic 6, Slide 21

22 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) Logic: Datalog CPSC 322 Logic 6, Slide 21

23 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) noisy(pencil) Logic: Datalog CPSC 322 Logic 6, Slide 21

24 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) noisy(pencil) left of(phone, pencil) false Logic: Datalog CPSC 322 Logic 6, Slide 21

25 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) noisy(pencil) lef t of(phone, pencil) lef t of(phone, telephone) false Logic: Datalog CPSC 322 Logic 6, Slide 21

26 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) noisy(pencil) lef t of(phone, pencil) lef t of(phone, telephone) noisy(pencil) lef t of(phone, telephone) false false Logic: Datalog CPSC 322 Logic 6, Slide 21

27 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) noisy(pencil) lef t of(phone, pencil) lef t of(phone, telephone) noisy(pencil) lef t of(phone, telephone) noisy(pencil) lef t of(phone, pencil) false false Logic: Datalog CPSC 322 Logic 6, Slide 21

28 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) noisy(pencil) lef t of(phone, pencil) lef t of(phone, telephone) noisy(pencil) lef t of(phone, telephone) noisy(pencil) lef t of(phone, pencil) noisy(phone) noisy(telephone) noisy(pencil) false false false Logic: Datalog CPSC 322 Logic 6, Slide 21

29 Example Truths In the interpretation given before: noisy(phone) noisy(telephone) noisy(pencil) lef t of(phone, pencil) lef t of(phone, telephone) noisy(pencil) lef t of(phone, telephone) noisy(pencil) lef t of(phone, pencil) noisy(phone) noisy(telephone) noisy(pencil) false false false Logic: Datalog CPSC 322 Logic 6, Slide 21

30 Variables How do we determine the truth value of a clause that includes variables? Definition (variable assignment) A variable assignment is a function from variables into the domain. Given an interpretation and a variable assignment, each term denotes an individual and each clause is either or false. A clause containing variables is in an interpretation if it is for all variable assignments. Variables are universally quantified in the scope of a clause. Logic: Datalog CPSC 322 Logic 6, Slide 22

31 Models and logical consequences Definition (model) A model of a set of clauses is an interpretation in which all the clauses are. Definition (logical consequence) If KB is a set of clauses and g is a conjunction of atoms, g is a logical consequence of KB, written KB = g, if g is in every model of KB. That is, KB = g if there is no interpretation in which KB is and g is false. Logic: Datalog CPSC 322 Logic 6, Slide 23